Apple's Music Problem?
iTunes is the gateway to Apple's music, movies, TV, audiobooks, podcasts, and applications for your iOS devices.
Today, Apple Music, Apple's music streaming service, was released. You would think you could sign up for the service on iTunes, but it doesn't seem possible...

I would have expected to be displayed prominently in the front page.
Update: iTunes 12.2 is needed for Apple Music but is yet to be released.
But it turns out you need the iOS 8.4, which updates the OS to include Apple Music, to sign up for Apple Music. Turns out there were slight display issues with the Apple Music welcome screen...

(Credit: Hearthstoner)
Don't forget about Apple's iTunes Match which allows users to store their music library (up to 25,000 songs) on iCloud for $24.99/year. I don't know why anyone would pay for such a service when Google Music does this for free with double the limit (50,000), especially since Google supports iOS devices quite well.
Once Apple Music is integrated into iTunes completely, will Apple have a more unified product?
The brand "iTunes" has reached way beyond music as the name conveys. In the current system, iOS users need to go through iTunes for backing up their phone or even syncing their data. This forces iOS users to use iTunes to manage their device (including unlocking their devices after too many incorrect attempts). To be honest, many users probably don't even connect their devices to a computer, so they don't even care for iTunes (maybe even unaware it exists).
iTunes for Windows had been historically terrible (I haven't used it in a while), but it provided the means to get consumers using Windows to use their products and ecosystem (a significant reason the iPod was so big). Now, it remains to do just that.
Would Apple benefit from rebranding iTunes into something else? Would it be beneficial to detach the music aspect from iTunes and the rest of iTunes become something else? (Similar to something that is done on iOS) This could create a separation and would unload the complexities of iTunes. It may potentially lead into a deemphasis/isolation of music. Maybe it'll allow development teams focus more on their aspect of the platform. (Dare I say a SDK for iOS development on Windows?!)
Though, at the same time, too much change may confuse consumers.